// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_TRIANGULARMATRIX_H
#define EIGEN_TRIANGULARMATRIX_H

namespace Eigen {

namespace internal {

    template <int Side, typename TriangularType, typename Rhs> struct triangular_solve_retval;

}

/** \class TriangularBase
  * \ingroup Core_Module
  *
  * \brief Base class for triangular part in a matrix
  */
template <typename Derived> class TriangularBase : public EigenBase<Derived>
{
public:
    enum
    {
        Mode = internal::traits<Derived>::Mode,
        RowsAtCompileTime = internal::traits<Derived>::RowsAtCompileTime,
        ColsAtCompileTime = internal::traits<Derived>::ColsAtCompileTime,
        MaxRowsAtCompileTime = internal::traits<Derived>::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = internal::traits<Derived>::MaxColsAtCompileTime,

        SizeAtCompileTime = (internal::size_at_compile_time<internal::traits<Derived>::RowsAtCompileTime, internal::traits<Derived>::ColsAtCompileTime>::ret),
        /**< This is equal to the number of coefficients, i.e. the number of
          * rows times the number of columns, or to \a Dynamic if this is not
          * known at compile-time. \sa RowsAtCompileTime, ColsAtCompileTime */

        MaxSizeAtCompileTime =
            (internal::size_at_compile_time<internal::traits<Derived>::MaxRowsAtCompileTime, internal::traits<Derived>::MaxColsAtCompileTime>::ret)

    };
    typedef typename internal::traits<Derived>::Scalar Scalar;
    typedef typename internal::traits<Derived>::StorageKind StorageKind;
    typedef typename internal::traits<Derived>::StorageIndex StorageIndex;
    typedef typename internal::traits<Derived>::FullMatrixType DenseMatrixType;
    typedef DenseMatrixType DenseType;
    typedef Derived const& Nested;

    EIGEN_DEVICE_FUNC
    inline TriangularBase() { eigen_assert(!((int(Mode) & int(UnitDiag)) && (int(Mode) & int(ZeroDiag)))); }

    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index rows() const EIGEN_NOEXCEPT { return derived().rows(); }
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index cols() const EIGEN_NOEXCEPT { return derived().cols(); }
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index outerStride() const EIGEN_NOEXCEPT { return derived().outerStride(); }
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index innerStride() const EIGEN_NOEXCEPT { return derived().innerStride(); }

    // dummy resize function
    EIGEN_DEVICE_FUNC
    void resize(Index rows, Index cols)
    {
        EIGEN_UNUSED_VARIABLE(rows);
        EIGEN_UNUSED_VARIABLE(cols);
        eigen_assert(rows == this->rows() && cols == this->cols());
    }

    EIGEN_DEVICE_FUNC
    inline Scalar coeff(Index row, Index col) const { return derived().coeff(row, col); }
    EIGEN_DEVICE_FUNC
    inline Scalar& coeffRef(Index row, Index col) { return derived().coeffRef(row, col); }

    /** \see MatrixBase::copyCoeff(row,col)
      */
    template <typename Other> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void copyCoeff(Index row, Index col, Other& other)
    {
        derived().coeffRef(row, col) = other.coeff(row, col);
    }

    EIGEN_DEVICE_FUNC
    inline Scalar operator()(Index row, Index col) const
    {
        check_coordinates(row, col);
        return coeff(row, col);
    }
    EIGEN_DEVICE_FUNC
    inline Scalar& operator()(Index row, Index col)
    {
        check_coordinates(row, col);
        return coeffRef(row, col);
    }

#ifndef EIGEN_PARSED_BY_DOXYGEN
    EIGEN_DEVICE_FUNC
    inline const Derived& derived() const { return *static_cast<const Derived*>(this); }
    EIGEN_DEVICE_FUNC
    inline Derived& derived() { return *static_cast<Derived*>(this); }
#endif  // not EIGEN_PARSED_BY_DOXYGEN

    template <typename DenseDerived> EIGEN_DEVICE_FUNC void evalTo(MatrixBase<DenseDerived>& other) const;
    template <typename DenseDerived> EIGEN_DEVICE_FUNC void evalToLazy(MatrixBase<DenseDerived>& other) const;

    EIGEN_DEVICE_FUNC
    DenseMatrixType toDenseMatrix() const
    {
        DenseMatrixType res(rows(), cols());
        evalToLazy(res);
        return res;
    }

protected:
    void check_coordinates(Index row, Index col) const
    {
        EIGEN_ONLY_USED_FOR_DEBUG(row);
        EIGEN_ONLY_USED_FOR_DEBUG(col);
        eigen_assert(col >= 0 && col < cols() && row >= 0 && row < rows());
        const int mode = int(Mode) & ~SelfAdjoint;
        EIGEN_ONLY_USED_FOR_DEBUG(mode);
        eigen_assert((mode == Upper && col >= row) || (mode == Lower && col <= row) || ((mode == StrictlyUpper || mode == UnitUpper) && col > row) ||
                     ((mode == StrictlyLower || mode == UnitLower) && col < row));
    }

#ifdef EIGEN_INTERNAL_DEBUGGING
    void check_coordinates_internal(Index row, Index col) const { check_coordinates(row, col); }
#else
    void check_coordinates_internal(Index, Index) const {}
#endif
};

/** \class TriangularView
  * \ingroup Core_Module
  *
  * \brief Expression of a triangular part in a matrix
  *
  * \param MatrixType the type of the object in which we are taking the triangular part
  * \param Mode the kind of triangular matrix expression to construct. Can be #Upper,
  *             #Lower, #UnitUpper, #UnitLower, #StrictlyUpper, or #StrictlyLower.
  *             This is in fact a bit field; it must have either #Upper or #Lower,
  *             and additionally it may have #UnitDiag or #ZeroDiag or neither.
  *
  * This class represents a triangular part of a matrix, not necessarily square. Strictly speaking, for rectangular
  * matrices one should speak of "trapezoid" parts. This class is the return type
  * of MatrixBase::triangularView() and SparseMatrixBase::triangularView(), and most of the time this is the only way it is used.
  *
  * \sa MatrixBase::triangularView()
  */
namespace internal {
    template <typename MatrixType, unsigned int _Mode> struct traits<TriangularView<MatrixType, _Mode>> : traits<MatrixType>
    {
        typedef typename ref_selector<MatrixType>::non_const_type MatrixTypeNested;
        typedef typename remove_reference<MatrixTypeNested>::type MatrixTypeNestedNonRef;
        typedef typename remove_all<MatrixTypeNested>::type MatrixTypeNestedCleaned;
        typedef typename MatrixType::PlainObject FullMatrixType;
        typedef MatrixType ExpressionType;
        enum
        {
            Mode = _Mode,
            FlagsLvalueBit = is_lvalue<MatrixType>::value ? LvalueBit : 0,
            Flags = (MatrixTypeNestedCleaned::Flags & (HereditaryBits | FlagsLvalueBit) & (~(PacketAccessBit | DirectAccessBit | LinearAccessBit)))
        };
    };
}  // namespace internal

template <typename _MatrixType, unsigned int _Mode, typename StorageKind> class TriangularViewImpl;

template <typename _MatrixType, unsigned int _Mode>
class TriangularView : public TriangularViewImpl<_MatrixType, _Mode, typename internal::traits<_MatrixType>::StorageKind>
{
public:
    typedef TriangularViewImpl<_MatrixType, _Mode, typename internal::traits<_MatrixType>::StorageKind> Base;
    typedef typename internal::traits<TriangularView>::Scalar Scalar;
    typedef _MatrixType MatrixType;

protected:
    typedef typename internal::traits<TriangularView>::MatrixTypeNested MatrixTypeNested;
    typedef typename internal::traits<TriangularView>::MatrixTypeNestedNonRef MatrixTypeNestedNonRef;

    typedef typename internal::remove_all<typename MatrixType::ConjugateReturnType>::type MatrixConjugateReturnType;
    typedef TriangularView<typename internal::add_const<MatrixType>::type, _Mode> ConstTriangularView;

public:
    typedef typename internal::traits<TriangularView>::StorageKind StorageKind;
    typedef typename internal::traits<TriangularView>::MatrixTypeNestedCleaned NestedExpression;

    enum
    {
        Mode = _Mode,
        Flags = internal::traits<TriangularView>::Flags,
        TransposeMode = (Mode & Upper ? Lower : 0) | (Mode & Lower ? Upper : 0) | (Mode & (UnitDiag)) | (Mode & (ZeroDiag)),
        IsVectorAtCompileTime = false
    };

    EIGEN_DEVICE_FUNC
    explicit inline TriangularView(MatrixType& matrix) : m_matrix(matrix) {}

    EIGEN_INHERIT_ASSIGNMENT_OPERATORS(TriangularView)

    /** \copydoc EigenBase::rows() */
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index rows() const EIGEN_NOEXCEPT { return m_matrix.rows(); }
    /** \copydoc EigenBase::cols() */
    EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR inline Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }

    /** \returns a const reference to the nested expression */
    EIGEN_DEVICE_FUNC
    const NestedExpression& nestedExpression() const { return m_matrix; }

    /** \returns a reference to the nested expression */
    EIGEN_DEVICE_FUNC
    NestedExpression& nestedExpression() { return m_matrix; }

    typedef TriangularView<const MatrixConjugateReturnType, Mode> ConjugateReturnType;
    /** \sa MatrixBase::conjugate() const */
    EIGEN_DEVICE_FUNC
    inline const ConjugateReturnType conjugate() const { return ConjugateReturnType(m_matrix.conjugate()); }

    /** \returns an expression of the complex conjugate of \c *this if Cond==true,
     *           returns \c *this otherwise.
     */
    template <bool Cond> EIGEN_DEVICE_FUNC inline typename internal::conditional<Cond, ConjugateReturnType, ConstTriangularView>::type conjugateIf() const
    {
        typedef typename internal::conditional<Cond, ConjugateReturnType, ConstTriangularView>::type ReturnType;
        return ReturnType(m_matrix.template conjugateIf<Cond>());
    }

    typedef TriangularView<const typename MatrixType::AdjointReturnType, TransposeMode> AdjointReturnType;
    /** \sa MatrixBase::adjoint() const */
    EIGEN_DEVICE_FUNC
    inline const AdjointReturnType adjoint() const { return AdjointReturnType(m_matrix.adjoint()); }

    typedef TriangularView<typename MatrixType::TransposeReturnType, TransposeMode> TransposeReturnType;
    /** \sa MatrixBase::transpose() */
    EIGEN_DEVICE_FUNC
    inline TransposeReturnType transpose()
    {
        EIGEN_STATIC_ASSERT_LVALUE(MatrixType)
        typename MatrixType::TransposeReturnType tmp(m_matrix);
        return TransposeReturnType(tmp);
    }

    typedef TriangularView<const typename MatrixType::ConstTransposeReturnType, TransposeMode> ConstTransposeReturnType;
    /** \sa MatrixBase::transpose() const */
    EIGEN_DEVICE_FUNC
    inline const ConstTransposeReturnType transpose() const { return ConstTransposeReturnType(m_matrix.transpose()); }

    template <typename Other> EIGEN_DEVICE_FUNC inline const Solve<TriangularView, Other> solve(const MatrixBase<Other>& other) const
    {
        return Solve<TriangularView, Other>(*this, other.derived());
    }

// workaround MSVC ICE
#if EIGEN_COMP_MSVC
    template <int Side, typename Other>
    EIGEN_DEVICE_FUNC inline const internal::triangular_solve_retval<Side, TriangularView, Other> solve(const MatrixBase<Other>& other) const
    {
        return Base::template solve<Side>(other);
    }
#else
    using Base::solve;
#endif

    /** \returns a selfadjoint view of the referenced triangular part which must be either \c #Upper or \c #Lower.
      *
      * This is a shortcut for \code this->nestedExpression().selfadjointView<(*this)::Mode>() \endcode
      * \sa MatrixBase::selfadjointView() */
    EIGEN_DEVICE_FUNC
    SelfAdjointView<MatrixTypeNestedNonRef, Mode> selfadjointView()
    {
        EIGEN_STATIC_ASSERT((Mode & (UnitDiag | ZeroDiag)) == 0, PROGRAMMING_ERROR);
        return SelfAdjointView<MatrixTypeNestedNonRef, Mode>(m_matrix);
    }

    /** This is the const version of selfadjointView() */
    EIGEN_DEVICE_FUNC
    const SelfAdjointView<MatrixTypeNestedNonRef, Mode> selfadjointView() const
    {
        EIGEN_STATIC_ASSERT((Mode & (UnitDiag | ZeroDiag)) == 0, PROGRAMMING_ERROR);
        return SelfAdjointView<MatrixTypeNestedNonRef, Mode>(m_matrix);
    }

    /** \returns the determinant of the triangular matrix
      * \sa MatrixBase::determinant() */
    EIGEN_DEVICE_FUNC
    Scalar determinant() const
    {
        if (Mode & UnitDiag)
            return 1;
        else if (Mode & ZeroDiag)
            return 0;
        else
            return m_matrix.diagonal().prod();
    }

protected:
    MatrixTypeNested m_matrix;
};

/** \ingroup Core_Module
  *
  * \brief Base class for a triangular part in a \b dense matrix
  *
  * This class is an abstract base class of class TriangularView, and objects of type TriangularViewImpl cannot be instantiated.
  * It extends class TriangularView with additional methods which available for dense expressions only.
  *
  * \sa class TriangularView, MatrixBase::triangularView()
  */
template <typename _MatrixType, unsigned int _Mode>
class TriangularViewImpl<_MatrixType, _Mode, Dense> : public TriangularBase<TriangularView<_MatrixType, _Mode>>
{
public:
    typedef TriangularView<_MatrixType, _Mode> TriangularViewType;
    typedef TriangularBase<TriangularViewType> Base;
    typedef typename internal::traits<TriangularViewType>::Scalar Scalar;

    typedef _MatrixType MatrixType;
    typedef typename MatrixType::PlainObject DenseMatrixType;
    typedef DenseMatrixType PlainObject;

public:
    using Base::derived;
    using Base::evalToLazy;

    typedef typename internal::traits<TriangularViewType>::StorageKind StorageKind;

    enum
    {
        Mode = _Mode,
        Flags = internal::traits<TriangularViewType>::Flags
    };

    /** \returns the outer-stride of the underlying dense matrix
      * \sa DenseCoeffsBase::outerStride() */
    EIGEN_DEVICE_FUNC
    inline Index outerStride() const { return derived().nestedExpression().outerStride(); }
    /** \returns the inner-stride of the underlying dense matrix
      * \sa DenseCoeffsBase::innerStride() */
    EIGEN_DEVICE_FUNC
    inline Index innerStride() const { return derived().nestedExpression().innerStride(); }

    /** \sa MatrixBase::operator+=() */
    template <typename Other> EIGEN_DEVICE_FUNC TriangularViewType& operator+=(const DenseBase<Other>& other)
    {
        internal::call_assignment_no_alias(derived(), other.derived(), internal::add_assign_op<Scalar, typename Other::Scalar>());
        return derived();
    }
    /** \sa MatrixBase::operator-=() */
    template <typename Other> EIGEN_DEVICE_FUNC TriangularViewType& operator-=(const DenseBase<Other>& other)
    {
        internal::call_assignment_no_alias(derived(), other.derived(), internal::sub_assign_op<Scalar, typename Other::Scalar>());
        return derived();
    }

    /** \sa MatrixBase::operator*=() */
    EIGEN_DEVICE_FUNC
    TriangularViewType& operator*=(const typename internal::traits<MatrixType>::Scalar& other) { return *this = derived().nestedExpression() * other; }
    /** \sa DenseBase::operator/=() */
    EIGEN_DEVICE_FUNC
    TriangularViewType& operator/=(const typename internal::traits<MatrixType>::Scalar& other) { return *this = derived().nestedExpression() / other; }

    /** \sa MatrixBase::fill() */
    EIGEN_DEVICE_FUNC
    void fill(const Scalar& value) { setConstant(value); }
    /** \sa MatrixBase::setConstant() */
    EIGEN_DEVICE_FUNC
    TriangularViewType& setConstant(const Scalar& value) { return *this = MatrixType::Constant(derived().rows(), derived().cols(), value); }
    /** \sa MatrixBase::setZero() */
    EIGEN_DEVICE_FUNC
    TriangularViewType& setZero() { return setConstant(Scalar(0)); }
    /** \sa MatrixBase::setOnes() */
    EIGEN_DEVICE_FUNC
    TriangularViewType& setOnes() { return setConstant(Scalar(1)); }

    /** \sa MatrixBase::coeff()
      * \warning the coordinates must fit into the referenced triangular part
      */
    EIGEN_DEVICE_FUNC
    inline Scalar coeff(Index row, Index col) const
    {
        Base::check_coordinates_internal(row, col);
        return derived().nestedExpression().coeff(row, col);
    }

    /** \sa MatrixBase::coeffRef()
      * \warning the coordinates must fit into the referenced triangular part
      */
    EIGEN_DEVICE_FUNC
    inline Scalar& coeffRef(Index row, Index col)
    {
        EIGEN_STATIC_ASSERT_LVALUE(TriangularViewType);
        Base::check_coordinates_internal(row, col);
        return derived().nestedExpression().coeffRef(row, col);
    }

    /** Assigns a triangular matrix to a triangular part of a dense matrix */
    template <typename OtherDerived> EIGEN_DEVICE_FUNC TriangularViewType& operator=(const TriangularBase<OtherDerived>& other);

    /** Shortcut for\code *this = other.other.triangularView<(*this)::Mode>() \endcode */
    template <typename OtherDerived> EIGEN_DEVICE_FUNC TriangularViewType& operator=(const MatrixBase<OtherDerived>& other);

#ifndef EIGEN_PARSED_BY_DOXYGEN
    EIGEN_DEVICE_FUNC
    TriangularViewType& operator=(const TriangularViewImpl& other) { return *this = other.derived().nestedExpression(); }

    template <typename OtherDerived>
    /** \deprecated */
    EIGEN_DEPRECATED EIGEN_DEVICE_FUNC void lazyAssign(const TriangularBase<OtherDerived>& other);

    template <typename OtherDerived>
    /** \deprecated */
    EIGEN_DEPRECATED EIGEN_DEVICE_FUNC void lazyAssign(const MatrixBase<OtherDerived>& other);
#endif

    /** Efficient triangular matrix times vector/matrix product */
    template <typename OtherDerived> EIGEN_DEVICE_FUNC const Product<TriangularViewType, OtherDerived> operator*(const MatrixBase<OtherDerived>& rhs) const
    {
        return Product<TriangularViewType, OtherDerived>(derived(), rhs.derived());
    }

    /** Efficient vector/matrix times triangular matrix product */
    template <typename OtherDerived>
    friend EIGEN_DEVICE_FUNC const Product<OtherDerived, TriangularViewType> operator*(const MatrixBase<OtherDerived>& lhs, const TriangularViewImpl& rhs)
    {
        return Product<OtherDerived, TriangularViewType>(lhs.derived(), rhs.derived());
    }

    /** \returns the product of the inverse of \c *this with \a other, \a *this being triangular.
      *
      * This function computes the inverse-matrix matrix product inverse(\c *this) * \a other if
      * \a Side==OnTheLeft (the default), or the right-inverse-multiply  \a other * inverse(\c *this) if
      * \a Side==OnTheRight.
      *
      * Note that the template parameter \c Side can be omitted, in which case \c Side==OnTheLeft
      *
      * The matrix \c *this must be triangular and invertible (i.e., all the coefficients of the
      * diagonal must be non zero). It works as a forward (resp. backward) substitution if \c *this
      * is an upper (resp. lower) triangular matrix.
      *
      * Example: \include Triangular_solve.cpp
      * Output: \verbinclude Triangular_solve.out
      *
      * This function returns an expression of the inverse-multiply and can works in-place if it is assigned
      * to the same matrix or vector \a other.
      *
      * For users coming from BLAS, this function (and more specifically solveInPlace()) offer
      * all the operations supported by the \c *TRSV and \c *TRSM BLAS routines.
      *
      * \sa TriangularView::solveInPlace()
      */
    template <int Side, typename Other>
    inline const internal::triangular_solve_retval<Side, TriangularViewType, Other> solve(const MatrixBase<Other>& other) const;

    /** "in-place" version of TriangularView::solve() where the result is written in \a other
      *
      * \warning The parameter is only marked 'const' to make the C++ compiler accept a temporary expression here.
      * This function will const_cast it, so constness isn't honored here.
      *
      * Note that the template parameter \c Side can be omitted, in which case \c Side==OnTheLeft
      *
      * See TriangularView:solve() for the details.
      */
    template <int Side, typename OtherDerived> EIGEN_DEVICE_FUNC void solveInPlace(const MatrixBase<OtherDerived>& other) const;

    template <typename OtherDerived> EIGEN_DEVICE_FUNC void solveInPlace(const MatrixBase<OtherDerived>& other) const { return solveInPlace<OnTheLeft>(other); }

    /** Swaps the coefficients of the common triangular parts of two matrices */
    template <typename OtherDerived>
    EIGEN_DEVICE_FUNC
#ifdef EIGEN_PARSED_BY_DOXYGEN
        void
        swap(TriangularBase<OtherDerived>& other)
#else
        void
        swap(TriangularBase<OtherDerived> const& other)
#endif
    {
        EIGEN_STATIC_ASSERT_LVALUE(OtherDerived);
        call_assignment(derived(), other.const_cast_derived(), internal::swap_assign_op<Scalar>());
    }

    /** Shortcut for \code (*this).swap(other.triangularView<(*this)::Mode>()) \endcode */
    template <typename OtherDerived>
    /** \deprecated */
    EIGEN_DEPRECATED EIGEN_DEVICE_FUNC void swap(MatrixBase<OtherDerived> const& other)
    {
        EIGEN_STATIC_ASSERT_LVALUE(OtherDerived);
        call_assignment(derived(), other.const_cast_derived(), internal::swap_assign_op<Scalar>());
    }

    template <typename RhsType, typename DstType> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void _solve_impl(const RhsType& rhs, DstType& dst) const
    {
        if (!internal::is_same_dense(dst, rhs))
            dst = rhs;
        this->solveInPlace(dst);
    }

    template <typename ProductType>
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TriangularViewType& _assignProduct(const ProductType& prod, const Scalar& alpha, bool beta);

protected:
    EIGEN_DEFAULT_COPY_CONSTRUCTOR(TriangularViewImpl)
    EIGEN_DEFAULT_EMPTY_CONSTRUCTOR_AND_DESTRUCTOR(TriangularViewImpl)
};

/***************************************************************************
* Implementation of triangular evaluation/assignment
***************************************************************************/

#ifndef EIGEN_PARSED_BY_DOXYGEN
// FIXME should we keep that possibility
template <typename MatrixType, unsigned int Mode>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC inline TriangularView<MatrixType, Mode>& TriangularViewImpl<MatrixType, Mode, Dense>::operator=(const MatrixBase<OtherDerived>& other)
{
    internal::call_assignment_no_alias(derived(), other.derived(), internal::assign_op<Scalar, typename OtherDerived::Scalar>());
    return derived();
}

// FIXME should we keep that possibility
template <typename MatrixType, unsigned int Mode>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC void TriangularViewImpl<MatrixType, Mode, Dense>::lazyAssign(const MatrixBase<OtherDerived>& other)
{
    internal::call_assignment_no_alias(derived(), other.template triangularView<Mode>());
}

template <typename MatrixType, unsigned int Mode>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC inline TriangularView<MatrixType, Mode>& TriangularViewImpl<MatrixType, Mode, Dense>::operator=(const TriangularBase<OtherDerived>& other)
{
    eigen_assert(Mode == int(OtherDerived::Mode));
    internal::call_assignment(derived(), other.derived());
    return derived();
}

template <typename MatrixType, unsigned int Mode>
template <typename OtherDerived>
EIGEN_DEVICE_FUNC void TriangularViewImpl<MatrixType, Mode, Dense>::lazyAssign(const TriangularBase<OtherDerived>& other)
{
    eigen_assert(Mode == int(OtherDerived::Mode));
    internal::call_assignment_no_alias(derived(), other.derived());
}
#endif

/***************************************************************************
* Implementation of TriangularBase methods
***************************************************************************/

/** Assigns a triangular or selfadjoint matrix to a dense matrix.
  * If the matrix is triangular, the opposite part is set to zero. */
template <typename Derived> template <typename DenseDerived> EIGEN_DEVICE_FUNC void TriangularBase<Derived>::evalTo(MatrixBase<DenseDerived>& other) const
{
    evalToLazy(other.derived());
}

/***************************************************************************
* Implementation of TriangularView methods
***************************************************************************/

/***************************************************************************
* Implementation of MatrixBase methods
***************************************************************************/

/**
  * \returns an expression of a triangular view extracted from the current matrix
  *
  * The parameter \a Mode can have the following values: \c #Upper, \c #StrictlyUpper, \c #UnitUpper,
  * \c #Lower, \c #StrictlyLower, \c #UnitLower.
  *
  * Example: \include MatrixBase_triangularView.cpp
  * Output: \verbinclude MatrixBase_triangularView.out
  *
  * \sa class TriangularView
  */
template <typename Derived>
template <unsigned int Mode>
EIGEN_DEVICE_FUNC typename MatrixBase<Derived>::template TriangularViewReturnType<Mode>::Type MatrixBase<Derived>::triangularView()
{
    return typename TriangularViewReturnType<Mode>::Type(derived());
}

/** This is the const version of MatrixBase::triangularView() */
template <typename Derived>
template <unsigned int Mode>
EIGEN_DEVICE_FUNC typename MatrixBase<Derived>::template ConstTriangularViewReturnType<Mode>::Type MatrixBase<Derived>::triangularView() const
{
    return typename ConstTriangularViewReturnType<Mode>::Type(derived());
}

/** \returns true if *this is approximately equal to an upper triangular matrix,
  *          within the precision given by \a prec.
  *
  * \sa isLowerTriangular()
  */
template <typename Derived> bool MatrixBase<Derived>::isUpperTriangular(const RealScalar& prec) const
{
    RealScalar maxAbsOnUpperPart = static_cast<RealScalar>(-1);
    for (Index j = 0; j < cols(); ++j)
    {
        Index maxi = numext::mini(j, rows() - 1);
        for (Index i = 0; i <= maxi; ++i)
        {
            RealScalar absValue = numext::abs(coeff(i, j));
            if (absValue > maxAbsOnUpperPart)
                maxAbsOnUpperPart = absValue;
        }
    }
    RealScalar threshold = maxAbsOnUpperPart * prec;
    for (Index j = 0; j < cols(); ++j)
        for (Index i = j + 1; i < rows(); ++i)
            if (numext::abs(coeff(i, j)) > threshold)
                return false;
    return true;
}

/** \returns true if *this is approximately equal to a lower triangular matrix,
  *          within the precision given by \a prec.
  *
  * \sa isUpperTriangular()
  */
template <typename Derived> bool MatrixBase<Derived>::isLowerTriangular(const RealScalar& prec) const
{
    RealScalar maxAbsOnLowerPart = static_cast<RealScalar>(-1);
    for (Index j = 0; j < cols(); ++j)
        for (Index i = j; i < rows(); ++i)
        {
            RealScalar absValue = numext::abs(coeff(i, j));
            if (absValue > maxAbsOnLowerPart)
                maxAbsOnLowerPart = absValue;
        }
    RealScalar threshold = maxAbsOnLowerPart * prec;
    for (Index j = 1; j < cols(); ++j)
    {
        Index maxi = numext::mini(j, rows() - 1);
        for (Index i = 0; i < maxi; ++i)
            if (numext::abs(coeff(i, j)) > threshold)
                return false;
    }
    return true;
}

/***************************************************************************
****************************************************************************
* Evaluators and Assignment of triangular expressions
***************************************************************************
***************************************************************************/

namespace internal {

    // TODO currently a triangular expression has the form TriangularView<.,.>
    //      in the future triangular-ness should be defined by the expression traits
    //      such that Transpose<TriangularView<.,.> > is valid. (currently TriangularBase::transpose() is overloaded to make it work)
    template <typename MatrixType, unsigned int Mode> struct evaluator_traits<TriangularView<MatrixType, Mode>>
    {
        typedef typename storage_kind_to_evaluator_kind<typename MatrixType::StorageKind>::Kind Kind;
        typedef typename glue_shapes<typename evaluator_traits<MatrixType>::Shape, TriangularShape>::type Shape;
    };

    template <typename MatrixType, unsigned int Mode>
    struct unary_evaluator<TriangularView<MatrixType, Mode>, IndexBased> : evaluator<typename internal::remove_all<MatrixType>::type>
    {
        typedef TriangularView<MatrixType, Mode> XprType;
        typedef evaluator<typename internal::remove_all<MatrixType>::type> Base;
        EIGEN_DEVICE_FUNC
        unary_evaluator(const XprType& xpr) : Base(xpr.nestedExpression()) {}
    };

    // Additional assignment kinds:
    struct Triangular2Triangular
    {
    };
    struct Triangular2Dense
    {
    };
    struct Dense2Triangular
    {
    };

    template <typename Kernel, unsigned int Mode, int UnrollCount, bool ClearOpposite> struct triangular_assignment_loop;

    /** \internal Specialization of the dense assignment kernel for triangular matrices.
  * The main difference is that the triangular, diagonal, and opposite parts are processed through three different functions.
  * \tparam UpLo must be either Lower or Upper
  * \tparam Mode must be either 0, UnitDiag, ZeroDiag, or SelfAdjoint
  */
    template <int UpLo, int Mode, int SetOpposite, typename DstEvaluatorTypeT, typename SrcEvaluatorTypeT, typename Functor, int Version = Specialized>
    class triangular_dense_assignment_kernel : public generic_dense_assignment_kernel<DstEvaluatorTypeT, SrcEvaluatorTypeT, Functor, Version>
    {
    protected:
        typedef generic_dense_assignment_kernel<DstEvaluatorTypeT, SrcEvaluatorTypeT, Functor, Version> Base;
        typedef typename Base::DstXprType DstXprType;
        typedef typename Base::SrcXprType SrcXprType;
        using Base::m_dst;
        using Base::m_functor;
        using Base::m_src;

    public:
        typedef typename Base::DstEvaluatorType DstEvaluatorType;
        typedef typename Base::SrcEvaluatorType SrcEvaluatorType;
        typedef typename Base::Scalar Scalar;
        typedef typename Base::AssignmentTraits AssignmentTraits;

        EIGEN_DEVICE_FUNC triangular_dense_assignment_kernel(DstEvaluatorType& dst, const SrcEvaluatorType& src, const Functor& func, DstXprType& dstExpr)
            : Base(dst, src, func, dstExpr)
        {
        }

#ifdef EIGEN_INTERNAL_DEBUGGING
        EIGEN_DEVICE_FUNC void assignCoeff(Index row, Index col)
        {
            eigen_internal_assert(row != col);
            Base::assignCoeff(row, col);
        }
#else
        using Base::assignCoeff;
#endif

        EIGEN_DEVICE_FUNC void assignDiagonalCoeff(Index id)
        {
            if (Mode == UnitDiag && SetOpposite)
                m_functor.assignCoeff(m_dst.coeffRef(id, id), Scalar(1));
            else if (Mode == ZeroDiag && SetOpposite)
                m_functor.assignCoeff(m_dst.coeffRef(id, id), Scalar(0));
            else if (Mode == 0)
                Base::assignCoeff(id, id);
        }

        EIGEN_DEVICE_FUNC void assignOppositeCoeff(Index row, Index col)
        {
            eigen_internal_assert(row != col);
            if (SetOpposite)
                m_functor.assignCoeff(m_dst.coeffRef(row, col), Scalar(0));
        }
    };

    template <int Mode, bool SetOpposite, typename DstXprType, typename SrcXprType, typename Functor>
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void call_triangular_assignment_loop(DstXprType& dst, const SrcXprType& src, const Functor& func)
    {
        typedef evaluator<DstXprType> DstEvaluatorType;
        typedef evaluator<SrcXprType> SrcEvaluatorType;

        SrcEvaluatorType srcEvaluator(src);

        Index dstRows = src.rows();
        Index dstCols = src.cols();
        if ((dst.rows() != dstRows) || (dst.cols() != dstCols))
            dst.resize(dstRows, dstCols);
        DstEvaluatorType dstEvaluator(dst);

        typedef triangular_dense_assignment_kernel<Mode&(Lower | Upper),
                                                   Mode&(UnitDiag | ZeroDiag | SelfAdjoint),
                                                   SetOpposite,
                                                   DstEvaluatorType,
                                                   SrcEvaluatorType,
                                                   Functor>
            Kernel;
        Kernel kernel(dstEvaluator, srcEvaluator, func, dst.const_cast_derived());

        enum
        {
            unroll = DstXprType::SizeAtCompileTime != Dynamic && SrcEvaluatorType::CoeffReadCost < HugeCost &&
                     DstXprType::SizeAtCompileTime * (int(DstEvaluatorType::CoeffReadCost) + int(SrcEvaluatorType::CoeffReadCost)) / 2 <= EIGEN_UNROLLING_LIMIT
        };

        triangular_assignment_loop<Kernel, Mode, unroll ? int(DstXprType::SizeAtCompileTime) : Dynamic, SetOpposite>::run(kernel);
    }

    template <int Mode, bool SetOpposite, typename DstXprType, typename SrcXprType>
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void call_triangular_assignment_loop(DstXprType& dst, const SrcXprType& src)
    {
        call_triangular_assignment_loop<Mode, SetOpposite>(dst, src, internal::assign_op<typename DstXprType::Scalar, typename SrcXprType::Scalar>());
    }

    template <> struct AssignmentKind<TriangularShape, TriangularShape>
    {
        typedef Triangular2Triangular Kind;
    };
    template <> struct AssignmentKind<DenseShape, TriangularShape>
    {
        typedef Triangular2Dense Kind;
    };
    template <> struct AssignmentKind<TriangularShape, DenseShape>
    {
        typedef Dense2Triangular Kind;
    };

    template <typename DstXprType, typename SrcXprType, typename Functor> struct Assignment<DstXprType, SrcXprType, Functor, Triangular2Triangular>
    {
        EIGEN_DEVICE_FUNC static void run(DstXprType& dst, const SrcXprType& src, const Functor& func)
        {
            eigen_assert(int(DstXprType::Mode) == int(SrcXprType::Mode));

            call_triangular_assignment_loop<DstXprType::Mode, false>(dst, src, func);
        }
    };

    template <typename DstXprType, typename SrcXprType, typename Functor> struct Assignment<DstXprType, SrcXprType, Functor, Triangular2Dense>
    {
        EIGEN_DEVICE_FUNC static void run(DstXprType& dst, const SrcXprType& src, const Functor& func)
        {
            call_triangular_assignment_loop<SrcXprType::Mode, (int(SrcXprType::Mode) & int(SelfAdjoint)) == 0>(dst, src, func);
        }
    };

    template <typename DstXprType, typename SrcXprType, typename Functor> struct Assignment<DstXprType, SrcXprType, Functor, Dense2Triangular>
    {
        EIGEN_DEVICE_FUNC static void run(DstXprType& dst, const SrcXprType& src, const Functor& func)
        {
            call_triangular_assignment_loop<DstXprType::Mode, false>(dst, src, func);
        }
    };

    template <typename Kernel, unsigned int Mode, int UnrollCount, bool SetOpposite> struct triangular_assignment_loop
    {
        // FIXME: this is not very clean, perhaps this information should be provided by the kernel?
        typedef typename Kernel::DstEvaluatorType DstEvaluatorType;
        typedef typename DstEvaluatorType::XprType DstXprType;

        enum
        {
            col = (UnrollCount - 1) / DstXprType::RowsAtCompileTime,
            row = (UnrollCount - 1) % DstXprType::RowsAtCompileTime
        };

        typedef typename Kernel::Scalar Scalar;

        EIGEN_DEVICE_FUNC
        static inline void run(Kernel& kernel)
        {
            triangular_assignment_loop<Kernel, Mode, UnrollCount - 1, SetOpposite>::run(kernel);

            if (row == col)
                kernel.assignDiagonalCoeff(row);
            else if (((Mode & Lower) && row > col) || ((Mode & Upper) && row < col))
                kernel.assignCoeff(row, col);
            else if (SetOpposite)
                kernel.assignOppositeCoeff(row, col);
        }
    };

    // prevent buggy user code from causing an infinite recursion
    template <typename Kernel, unsigned int Mode, bool SetOpposite> struct triangular_assignment_loop<Kernel, Mode, 0, SetOpposite>
    {
        EIGEN_DEVICE_FUNC
        static inline void run(Kernel&) {}
    };

    // TODO: experiment with a recursive assignment procedure splitting the current
    //       triangular part into one rectangular and two triangular parts.

    template <typename Kernel, unsigned int Mode, bool SetOpposite> struct triangular_assignment_loop<Kernel, Mode, Dynamic, SetOpposite>
    {
        typedef typename Kernel::Scalar Scalar;
        EIGEN_DEVICE_FUNC
        static inline void run(Kernel& kernel)
        {
            for (Index j = 0; j < kernel.cols(); ++j)
            {
                Index maxi = numext::mini(j, kernel.rows());
                Index i = 0;
                if (((Mode & Lower) && SetOpposite) || (Mode & Upper))
                {
                    for (; i < maxi; ++i)
                        if (Mode & Upper)
                            kernel.assignCoeff(i, j);
                        else
                            kernel.assignOppositeCoeff(i, j);
                }
                else
                    i = maxi;

                if (i < kernel.rows())  // then i==j
                    kernel.assignDiagonalCoeff(i++);

                if (((Mode & Upper) && SetOpposite) || (Mode & Lower))
                {
                    for (; i < kernel.rows(); ++i)
                        if (Mode & Lower)
                            kernel.assignCoeff(i, j);
                        else
                            kernel.assignOppositeCoeff(i, j);
                }
            }
        }
    };

}  // end namespace internal

/** Assigns a triangular or selfadjoint matrix to a dense matrix.
  * If the matrix is triangular, the opposite part is set to zero. */
template <typename Derived> template <typename DenseDerived> EIGEN_DEVICE_FUNC void TriangularBase<Derived>::evalToLazy(MatrixBase<DenseDerived>& other) const
{
    other.derived().resize(this->rows(), this->cols());
    internal::call_triangular_assignment_loop<Derived::Mode, (int(Derived::Mode) & int(SelfAdjoint)) == 0 /* SetOpposite */>(other.derived(),
                                                                                                                             derived().nestedExpression());
}

namespace internal {

    // Triangular = Product
    template <typename DstXprType, typename Lhs, typename Rhs, typename Scalar>
    struct Assignment<DstXprType,
                      Product<Lhs, Rhs, DefaultProduct>,
                      internal::assign_op<Scalar, typename Product<Lhs, Rhs, DefaultProduct>::Scalar>,
                      Dense2Triangular>
    {
        typedef Product<Lhs, Rhs, DefaultProduct> SrcXprType;
        static void run(DstXprType& dst, const SrcXprType& src, const internal::assign_op<Scalar, typename SrcXprType::Scalar>&)
        {
            Index dstRows = src.rows();
            Index dstCols = src.cols();
            if ((dst.rows() != dstRows) || (dst.cols() != dstCols))
                dst.resize(dstRows, dstCols);

            dst._assignProduct(src, Scalar(1), false);
        }
    };

    // Triangular += Product
    template <typename DstXprType, typename Lhs, typename Rhs, typename Scalar>
    struct Assignment<DstXprType,
                      Product<Lhs, Rhs, DefaultProduct>,
                      internal::add_assign_op<Scalar, typename Product<Lhs, Rhs, DefaultProduct>::Scalar>,
                      Dense2Triangular>
    {
        typedef Product<Lhs, Rhs, DefaultProduct> SrcXprType;
        static void run(DstXprType& dst, const SrcXprType& src, const internal::add_assign_op<Scalar, typename SrcXprType::Scalar>&)
        {
            dst._assignProduct(src, Scalar(1), true);
        }
    };

    // Triangular -= Product
    template <typename DstXprType, typename Lhs, typename Rhs, typename Scalar>
    struct Assignment<DstXprType,
                      Product<Lhs, Rhs, DefaultProduct>,
                      internal::sub_assign_op<Scalar, typename Product<Lhs, Rhs, DefaultProduct>::Scalar>,
                      Dense2Triangular>
    {
        typedef Product<Lhs, Rhs, DefaultProduct> SrcXprType;
        static void run(DstXprType& dst, const SrcXprType& src, const internal::sub_assign_op<Scalar, typename SrcXprType::Scalar>&)
        {
            dst._assignProduct(src, Scalar(-1), true);
        }
    };

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_TRIANGULARMATRIX_H
